Adjugate of a matrix

It’s the transpose of The Cofactor Matrix.

2x2

Switch one of the diagonals, multiply the other with -1.

$$\begin{array}{r}A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\\ \\ \text{adj}A=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\end{array}$$

Why care?

From Theorem 36.1 (in reference):

$$\begin{array}{r}A(\text{adj}A)=(\text{det }A)I=(\text{adj }A)A\end{array}$$

This is similar to Matrix Inverses.

• Inverses return $I$ when multiplied with.
• adjugate returns a scaled $I$ when multiplied with.

Still, why care?

Alt formula for inverse! If determinant != 0, then:

$$A^{-1}=\frac{1}{\text{det }A}\text{ adj }A$$